*Yeah, yeah. Greek, Latin, who cares?

Thursday, September 9, 2010

Energy Schmenergy—what about FOOD?
(Prey Choice, Diet Breadth, and all that jazz)

For those not familiar with the use of optimal foraging models in archaeology, there’s this thing called the prey-choice model, or sometimes the diet-breadth model, that zooarchaeologists like to use in the interpretation of faunal assemblages. Originally developed by ecologists and typically presented in an evolutionary ecology framework, the model basically tells you which food resources an organism should exploit and which it should not if you’re willing to assume the organism is maximizing food-acquisition efficiency. More specifically, it focuses on maximizing the net rate of energetic gain (nice mouthful, huh?).

The prey-choice/diet-breadth model is formulated as an inequality. When true, the resource j should be pursued on encounter; otherwise, it should be bypassed:

The most important thing to bear in mind here is that the resources are ordered by their ei/hi ratios. Resource #1 (i=1) has the highest ei/hi ratio, resource #2 (i=2) has the next highest, etc. With that in mind, what are all these silly letters?

  • ei is the net energetic return of resource i (that is, the energy obtained from consuming the resource minus the about of energy expended in acquiring, processing (if applicable), and consuming the resource)
  • hi is the average handling time of resource i (that is, the amount of time required to obtain the resource once it has been encountered)
  • Ts is the time spent searching for resources to exploit
  • λi is the encounter rate with resource i (how often per unit time the resource is chanced upon)
  • s is the energetic cost (energy expended per unit time) of searching

The model subtracts the calories spent by the forager in acquiring the resource from the forager’s caloric gain from eating the resource and then divides that by the amount of time involved. This “net rate of energetic gain” (the left side of the inequality, and the value on which the resources are ordered – “ranked”) is compared to the overall net rate of energetic gain that would be expected if the forager only exploited more efficient—higher ranked—resources (the right side of the inequality). If the drop in efficiency caused by pursuing a less efficient resource would be outweighed by the efficiency cost of waiting for a more efficient resource to be found, then that less-efficient resource should be exploited and is part of the optimal diet. (Yes, I know that’s kind of confusing if you’re not familiar with it.)

The key point for most zooarchaeological uses is that resources are in or out of the optimal diet depending on the rate at which the more efficient resources are encountered (that is, how long one must search for the ‘better’ resource and how much energy would be expended in the process are the critical factors). Zooarchaeologists commonly use this to interpret faunal assemblages by looking for the addition (more usually, the increased representation) of what are thought to be lower-ranked (less efficient) resources and interpreting that as indicating a reduction in the availability of higher-ranked resources. Some attention is paid to whether or not there might be some environmental change that resulted in this reduction (or, if the lower-ranked resource did not appear, but simply increased in frequency some such change that resulted in increased availability of the lower-ranked resource). Finding no evidence of such environmental change, the inferred reduction in the availability of the higher-ranked resource is attributed to human agency, usually human population growth and associated overhunting of the most efficient resources. I don’t want to into the question of whether or not that logic chain is acceptable here...I’ve got a different axe to grind today:

If you stop and think about it, there’s a problem when it comes to hunting of medium to large animals, like most ungulates: the individual hunter almost certainly can’t eat all of the meat him/herself. And even if it were actually possible for the hunter to do so, because of sufficient ability to store the meat (say, a big freezer at home in the garage), he/she probably won’t actually eat all the meat. Rather, a lot of it—almost certainly a majority—will be shared with others. But what does this mean for the prey-choice model? Shouldn’t we only be including the meat the forager actually ate when we calculate ei? After all, he/she doesn’t really get any energetic benefit from the meat eaten by others (certainly not directly enough for it to be considered in determining the net rate of energetic return from the resource). But in that case, why is the forager going after these big animals so often, as is so frequently the case in, for example, the Middle Paleolithic? (Sure, the model could be inoperative...but we're assuming that at least something similar is going on.) There are some fairly easy answers to that question, such as the showing-off hypothesis or reciprocity with others doing the same thing, but we’re supposed to be using the prey-choice model here, which is silent on these topics.

What is to be done? Well, why not think about a slightly different formulation of the prey-choice model, one which fits this sort of behavior better, and in fact seems to match up better with the way archaeologists actually apply the model? Instead of maximizing the forager’s personal net energetic return rate, we’ll try maximizing the forager’s meat acquisition rate (we’re restricting ourselves to hunting here) . In doing so, we are implicitly (well, I guess it’s explicit now that I’m talking about it) assuming that the value of meat actually consumed by the forager and meat acquired and shared with others are the same. In cases where personal survival is at stake, this obviously isn’t likely to be the case, but it should be a reasonable approximation in a reciprocity situation and not too unreasonable—I hope—in a prestige situation. If nothing else, it should be a better fit for reciprocity or prestige than calories are!

Math warning!!! (Skip to here if you’re willing to take my word for the math.) This modification of the model involves replacing the net energetic return with the raw meat yield (there is no meat cost, so we’re no longer talking about a “net” value) and removing the subtraction of energetic cost of search from the right-side numerator, since we are only worried about the time, not the energy, expended in searching for prey. The revised equation looks like this:

Again, resources are ranked in order from highest to lowest ratios of food yield to handling time (yi/hi) so that all resources i such that i < j are higher-ranked than resource j. yi is the meat yield per engagement (encounter and pursuit) value, replacing ei, the net energetic gain per engagement value. Other terms are as listed previously. One really nice thing about this formulation is that the lack of the energetic cost of search factor in the numerator means that it can be simplified a lot more easily than the standard version. To do so, we first cancel out the search time terms:

Next, we define some substitutions:

defines an overall encounter rate with resources more highly ranked than resource j.

defines an encounter-rate-weighted average yield. Each higher-ranked resource’s yield is weighted by how often it is encountered. This can thus be thought of as the average (and thus expected) yield of the next encounter with a higher-ranked resource.

does the same thing for handling time. Once we have these terms defined, we can substitute them into the food-yield prey-choice model equation:

Dividing top and bottom of the right side by Λj converts this to:

This formulation makes it much more clear how the prey-choice model works. 1/Λj is simply the average time until the next encounter with a resource ranked higher than resource j. Thus, resource j should be pursued on encounter if its yield to handling time ratio is higher than the ratio of the expected yield of the next-encountered higher-ranked resource to the time required to first encounter and then handle that higher-ranked resource.

The standard prey-choice model works the same way, but with the complication of the energetic cost of search, the impact of which is hard to wrap one’s head around. As a general comparison, the food-yield version predicts (even if we assume that the consumption issues vis-à-vis energy discussed earlier are not operative) higher average efficiency of bypassing a given resource in favor of later encounters with higher-ranked ones (because the energy expended during search is not subtracted) and thus higher efficiency thresholds for the inclusion of lower-ranked resources. Meaning: the food-yield version predicts a greater focus on larger resources.

More general benefits of the food-yield version of the prey-choice model include not only the conversion to more readily understood (and measured!) characteristics of resources and foragers, but also a renewed emphasis on terms other than encounter rates as explanations for change. Neither yield nor handling time is necessarily a constant attribute of a resource, topics I will return to in the future.

NOTE: This is an informally written “zero-th” draft of something I’ve been messing with for some time. I have a couple of more application-oriented issues (alluded to in that last sentence) in mind that develop from this formulation of the prey-choice model...but I have been unable so far to effectively cram the model (re)development in with the substance of either one of those issues. What I’m mostly looking for here is any feedback on whether or not a formalized version of this would work as a standalone article (that is, much as it appears here, without any fleshed-out applications.

No comments:

Post a Comment